What is the difference between a stationary and a non-stationary policy?
A stationary policy, $\pi_t$, is a policy that does not change over time, that is, $\pi_t = \pi, \forall t \geq 0$, where $\pi$ can either be a function, $\pi: S \rightarrow A$ (a deterministic policy), or a conditional density, $\pi(A \mid S)$ (a stochastic policy). A non-stationary policy is a policy that is not stationary. More precisely, $\pi_i$ may not be equal to $\pi_j$, for $i \neq j \geq 0$, where $i$ and $j$ are thus two different time steps.
There are problems where a stationary optimal policy is guaranteed to exist. For example, in the case of a stochastic (there is a probability density that models the dynamics of the environment, that is, the transition function and the reward function) and discrete-time Markov decision process (MDP) with finite numbers of states and actions, and bounded rewards, where the objective is the long-run average reward, a stationary optimal policy exists. The proof of this fact is in the book Markov Decision Processes: Discrete Stochastic Dynamic Programming (1994), by Martin L. Puterman, which apparently is not freely available on the web.